Eigenmodes and resonance vibrations of 2D nanomembranes -- Graphene and hexagonal boron-nitride
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Eigenmodes and resonance vibrations of 2D nanomembranes – Graphene andhexagonal boron-nitride
Alexander V. Savin N.N. Semenov Federal Research Center for Chemical Physics,Russian Academy of Sciences (FRCCP RAS), Moscow, 119991, Russia
Natural and resonant oscillations of suspended circular graphene and hexagonal boron nitride (h-BN) membranes (single-layer sheets lying on a flat substrate having a circular hole of radius R ) havebeen simulated using full-atomic models. Substrates formed by flat surfaces of graphite and h-BNcrystal, hexagonal ice, silicon carbide 6H-SiC and nickel surface (111) have been used. The presenceof the substrate leads to the forming of a gap at the bottom of the frequency spectrum of transversalvibrations of the sheet. The frequencies of natural oscillations of the membrane (oscillations localizedon the suspended section of the sheet) always lie in this gap, and the frequencies of oscillationsdecrease by increasing radius of the membrane as ( R + R i ) − with nonezero effective increase ofradius R i >
0. The modeling of the sheet dynamics has shown that small periodic transversaldisplacements of the substrate lead to resonant vibrations of the membranes at frequencies closeto eigenfrequencies of nodeless vibrations of membranes with a circular symmetry. The energydistribution of resonant vibrations of the membrane has a circular symmetry and several nodalcircles, whose number i coincides with the number of the resonant frequency. The frequenciesof the resonances decrease by increasing the radius of the membrane as ( R + R i ) α i with exponent α i <
2. The lower rate of resonance frequency decrease is caused by the anharmonicity of membranevibrations.
I. INTRODUCTION
Being a nanosized polymorph of carbon, graphene at-tracts increased attention of researchers due to its uniquephysical properties [1, 2]. The remarkable propertiesof graphene have enabled the exploitation of graphenefor the development of nano-electro-mechanical system(NEMS) such as nanoresonators [3, 4]. The vibrationalproperties of graphene play an important role in analy-sis and design of graphene-based sensors and resonators.The aim of this work is to simulate the eigenmodes andresonant vibrations of suspended circular graphene (G)and hexagonal boron nitride (h-BN) membranes. Such2D membranes are formed as single-layer G and h-BNsheets lying on a flat substrate with a circular hole – seeFig. 1. These one atom thick membranes can be usedas highly efficient nanomechanical resonators [5–9] andas extraordinary sensitive detectors of mass, force andpressure [10–12].For analysis of vibrations of such membranes contin-uum models in which a sheet of graphene is consideredas continuous thin plate or thin shell [13–16] are usu-ally used. In this paper we will use discrete (full-atomic)models that take into account the hexagonal structure ofthe sheets. As substrate, we consider the plane surfacesof an ideal graphite and h-BN crystals, hexagonal ice I h ,silicon carbide 6H-SiC and the surface (111) of Nickelcrystal. II. MODEL
To calculate the interaction energy of a sheet (grapheneand h-BN) with a flat substrate the sheet has been placedparallel to the substrate surface. The interaction poten- tial of each atom belonging to the sheet with the sub-strate W ( h ) can be found as the function of the distanceto the substrate plane h , as a sum of its interaction en-ergies with the substrate atoms. The interaction of pairsof atoms has been described by the Lennard-Jones (LJ)potential (6,12): V LJ ( r ) = ǫ [( r /r ) − r /r ) ] , (1)where ǫ is the binding energy and r is the bond length.To find the interaction energy of graphene with the crys-talline graphite surface, we used the potential parameterstaken from [17] and, for other substrates, from [18]. Ta-ble I shows the parameters of LJ potential (1) for variousatomic pairs.The calculations have been made for the 2.0 × graphene (h-BN) sheet consisting of 160 carbon (boronand nitride) atoms, which is arranged in parallel to thecrystal surface at distance h . At each value of distance h , the energy was averaged over the shifts along sub-strate surface and, then, normalized on the number ofatoms in the graphene (h-BN) sheet. As a result, we ob-tained the dependence of the interaction energy of oneatom of the sheet with the substrate on its distance fromsubstrate plane W ( h ). The calculations showed that theinteraction energy with the substrate W ( h ) can be de-scribed with a high accuracy by the Lennard-Jones po-tential ( k, l ): W ( h ) = ǫ [ k ( h /h ) l − l ( h /h ) k ] / ( l − k ) , (2)where power l > k . Potential (2) has the minimum W ( h ) = − ǫ ( ǫ is the binding energy of the atom withsubstrate). The stiffness of interaction with the substrateis K = W ′′ ( h ) = ǫ lk/h . Table II presents the param-eters of LJ potential (2) for graphene and h-BN sheet onvarious substrates. Table I: Parameters of the LJ potential (1) for various pairs of interacting atoms.[17] CC [18] CC CH CO CSi NO NH NC NSi BO BH BC BSi ǫ (meV) 2.76 4.56 2.95 3.44 8.92 2.78 2.38 3.69 7.24 4.51 3.86 5.94 11.66 r (˚A) 3.809 3.851 3.369 3.676 4.073 3.579 3.25 3.754 3.965 3.78 3.433 3.965 4.188Figure 1: The rectangular sheet of graphene lying on a flatsubstrate having a circular hole in its center. The sheet con-sists of 9470 carbon atoms and has the shape of a rectangle ofsize 15 . × . . The radius of the hole in the substrate R = 6 nm.Table II: Parameters of ( k, l ) LJ potential (2) for grapheneand h-BN sheets on various substrates. ǫ (eV) h (˚A) l k K (N/m)Graphene – Ice I h h When graphene is located on the surface (111) of crys-talline nickel, a stronger chemical interaction of carbonatoms with the atoms of the substrate occurs (hybridiza-tion of the metal d -band with graphene π -states andcharge transfer from the metal to graphene). As a re-sult of the interaction of a graphene sheet with a crystalsurface a gap of the magnitude ω = 240 cm − appearsat the bottom of the frequency spectrum of transversaloscillations of the sheet [19]. From this we can estimate the harmonic coupling parameter of the interaction of thesheet atom with the substrate K = ω M C = 41 N/m( M C is the mass of carbon atom). Therefore, for smalldisplacements, the interaction with the substrate can bedescribed by the harmonic potential W ( h ) = 12 K ( h − h ) , (3)with stiffness coefficient K = 41 N/m and equilibriumdistance to the substrate plane h = 2 .
145 ˚A [20].To describe oscillations of the graphene and h-BNsheet, we present the system Hamiltonian in the form, H = N X n =1 (cid:20) M n ( ˙ u n , ˙ u n ) + P n + δ n W ( z n ) (cid:21) , (4)where M n is the mass of n -th atom of the sheet, u n =( x n ( t ) , y n ( t ) , z n ( t )) is the radius-vector of n -th atom atthe time t . The term P n describes the energy of inter-action of the atom with index n with the neighboringatoms, term W ( z n ) – the energy of interaction of theatom with substrate surface (the plane of the substratecoincides with the plane xy ). Coefficient δ n = 1 if n -thatom interacts and δ n = 0 if it does not interact with thesubstrate (if it lies above the hole in the substrate).To describe the dynamics of a graphene sheet, we usedthe interaction potentials described in detail in [21, 22],whereas to describe monolayer hexagonal boron nitride(h-BN) sheet we used extended Tersoff potential [23].We consider a hydrogen-terminated graphene (h-BN)sheet, where edge atoms correspond to the moleculargroup CH (BH or NH). We consider such a group as a sin-gle effective particle at the location of the carbon atom.Therefore, in our model of graphene nanoribbons we takethe mass of atoms inside the stripe as M n = 12 m p , andfor the edge atoms we consider a larger mass M n = 13 m p (where m p = 1 . · − kg is the proton mass).If we want to simulate the absence of a substrate fora part of the sheet atoms, we must take δ n = 0 for theseatoms. Figure 1 shows a square sheet of graphene of size15 . × . consisting of N = 9470 carbon atoms.The central circular part of the sheet does not interactwith the substrate (for atoms from this part the coef-ficient δ n = 0), forming a circular membrane of radius R = 6 nm. A similar structure was used to model thevibrations of a circular membrane made of h-BN sheet. ω ( c m − ) R (˚A) Figure 2: Dependence of frequencies ω of intrinsic transver-sal oscillations of a circular membrane of a graphene sheetlying on the surface of crystalline h-BN substrate on the ra-dius of the membrane R . Curves i = 1 , , , ω i = c i / ( R + R i ) , where coefficients c i = 3950,15500, 34500, 62500 cm − ˚A , additional radius R i = 4 .
0, 4.7,5.0, 5.7 ˚A. Straight line 5 shows the minimum frequency oftransversal oscillations of a graphene sheet on a flat substrate ω = ω = 78 .
87 cm − . Large markers show natural frequen-cies of intrinsic nodeless oscillations of the membrane withcircular symmetry. III. TRANSVERSAL NORMAL MODES
Let us consider the transversal vibrations of the atomsof the sheet. The natural frequencies and normal modeswere derived numerically as the solution of the problemon eigenvalues for matrices of the second derivatives ofsize N × N .When only transversal offsets are taken into account,the Hamiltonian of the sheet (4) can be written in theform H = 12 ( M ˙ Z , ˙ Z ) + P ( Z ) , (5)where M is a diagonal matrix of all masses of the sheet, Z = { z n − h } Nn =1 is N -dimensional vector of transversaldisplacements from equilibrium positions. Hamiltonian(5) corresponds to the motion equations, − M ¨ Z = ∂∂ Z P ( Z ) . (6)For small displacements, Eq. (6) reduces to a system oflinear equations, − M ¨ Z = BZ , (7)where the matrix has dimension N × N , B = (cid:18) ∂ P ∂z n ∂z n (cid:12)(cid:12)(cid:12)(cid:12) Z = (cid:19) N, Nn =1 , n =1 . Next, we make the transformation Z = M − / X ,and reduce the system (7) to the linear equations ofthe form − ¨ X = CX with the symmetric matrix C = M − / BM − / . Solutions of this linear system describethe eigenmodes of the sheet oscillations, which can bepresented in the form X ( t ) = A e exp( iωt ), where A isthe oscillation amplitude, ω = √ λ is the frequency, λ and e are the eigenvalue and normalized eigenvector ofthe matrix C [ C e = λ e , ( e , e ) = 1].The eigenvalues of the matrix C can be found numer-ically. Numerical matrix diagonalization demonstratesthat the presence of the substrate leads to the presenceof the gap [0 , ω ) at the bottom of the frequency spec-trum of transversal vibrations (minimum nonzero fre-quency ω = p K /M , for graphene M = M C ). Alleigen transversal vibrations of the sheet with frequencies ω < ω correspond to vibrations localized in the sus-pended central part of the sheet, i.e. to eigen vibrationsof the circular membrane.The dependence of the natural oscillations of the cir-cular membrane on its radius is shown in Fig. 2. Here,frequency ω = 78 .
87 cm − . A graphene membrane ona crystalline h-BN substrate with a radius of the centralhole in the substrate R = 5 has only one localized nat-ural oscillation, at R = 10 – 6 oscillations, at R = 15 –14, at R = 20 – 23, at R = 30 – 55, at R = 40 – 94,at R = 50 – 144 and at R = 60 ˚A – 209 natural oscil-lations. The minimum natural frequency of transversalvibrations of the membrane is approximated with highaccuracy by the dependence ω i ∼ c i / ( R + R i ) , (8)with index i = 1, c = 3950 cm − · ˚A , R = 4 . R i . The amount of additional magnifica-tion depends on the force of interaction with the sub-strate. The stronger is the interaction with the sub-strate, the smaller is the value of R . So R = 4 . R = 4 . R = 3 . R = 2 . i = 2, 3, ... – see Fig. 2. IV. ANHARMONISM OF MEMBRANEVIBRATIONS
To simulate the natural vibrations of the membrane,we must numerically integrate a system of equations ofmotion M n ¨ u n = − ∂H∂ u n , n = 1 , ..., N, (9) −4 −2 E ( e V ) ω (cm − ) Figure 3: Dependence of frequencies ω of the first and thesecond (curves 1 and 2) natural oscillations of the circularmembrane ( R = 6 nm) on energy of vibration E . Energy ofthermal vibrations E = k B T for temperature T = 300K isrepresented by a horizontal dashed line. with initial conditions x n (0) = x n , y n (0) = y n , , z n (0) = z n , ˙ x n (0) = 0 , ˙ y n (0) = 0 , ˙ z n (0) = Ae n , where { u n = ( x n , y n , z n ) } Nn =1 is the ground state of thegraphene sheet, e = { e n } Nn =1 is the eigenvector of thematrix C (amplitude A determines the energy of vibra-tions E = A ( Me , e ) / /t r with time relax-ation t r = 10 ps was introduced on edges of the sheet).Let us consider the dynamics of the natural vibrationof the membrane with radius R = 60 ˚A. Numerical in-tegration of the system of equations of motion (9) hasshown that when energy E < E = 0 .
01 eV the mem-brane performs harmonic oscillations with eigen modefrequency (frequency of the vibration ω does not de-pend on the energy E ). When E > E , the frequencyof the membrane vibration begins to increase monotoni-cally when increasing energy – see Fig. 3. Thus, at highenergy, the membrane behaves as anharmonic oscillatorwith rigid anharmonicity. Let us note that at room tem-perature T = 300K, the energy of thermal self-oscilattionof the membrane E > E ( E = k B T = 0 .
026 eV). There-fore, at room temperature, the thermal vibrations of thegraphene membrane will be anharmonic.
V. RESONANT VIBRATIONS
To analyze the resonant vibrations of a single-layermembrane, we will simulate the effect of periodictransversal changes in the position of the substrate onits vibrations. To do this, we numerically integrate thesystem of Langevin equations of motion M n ¨ x n = − ∂H∂x n + δ n [ − Γ M n ˙ x n + ξ n, ( t )] ,M n ¨ y n = − ∂H∂y n + δ n [ − Γ M n ˙ y n + ξ n, ( t )] , (10) ∆ T ( K ) (a) ∆ T ( K ) (b) ∆ T ( K ) ω (cm − ) (c) Figure 4: Dependence of additional thermalization of thegraphene membrane ∆ T on the oscillation frequency of theh-BN substrate ω for (a) membrane radius R = 2, (b) R = 4and (c) R = 6 nm. Red curves 1, 3, 5 show dependencies foramplitude of forced substrate vibrations A = 1, blue curves2, 4, 6 – dependencies for A = 2 ˚A · cm − . The circular mark-ers show the values of the frequencies of the intrinsic nodelessoscillations of the membrane with circular symmetry. M n ¨ z n = − ∂H∂z n + δ n [ − Γ M n ˙ z n + ξ n, ( t ) + F ( t, z n )] , where the coefficient δ n = 1 if the atom interacts withthe substrate and δ n = 0 if it does not interact (if it islocated in the suspended part of the sheet), Γ = 1 /t r is the friction coefficient, and random forces vectors( ξ n, , ξ n, , ξ n, ) are normalized as follows: h ξ n,i ( t ) ξ m,j ( t ) i = 2 M n Γ k B T δ nm δ ij δ ( t − t ) ,k B – Boltzmann constant, T – temperature of the ther-mostat, F – force of attraction of the atom to the sub-strate.If the position of the substrate plane is periodicallychanged along the z axis, then in the system of sheetmotion equations (10) the force F ( t, z ) = − W ′ ( z + Aω cos( ωt )) , where A and ω – the amplitude and the frequency of theforced oscillations of the substrate (by this definition theamplitude A characterizes the oscillation energy).Let us analyze at what frequencies of forced oscillationsof the substrate pumping of the energy to vibrations ofthe suspended section of the sheet will be the highest. ∆ T (K) ω (cm − ) R (nm) (b) ∆ T (K) ω (cm − ) R (nm) (c)
10 20 30 40 50 602 3 4 5 6 ∆ T (K) ω (cm − ) R (nm) (a) Figure 5: Dependence of additional thermalization ∆ T on theoscillation frequency of the substrate ω and on the membraneradius R for graphene membrane on substrates (a) Ni(111),(b) 6H-SiC(0001) and (c) for h-BN membrane on the sub-strate 6H-SiC(0001) (amplitude A = 2 ˚A · cm − ). For the sheet, the substrate is an external thermostat,so in the system of the equations of motion (10) onlyatoms in contact with the substrate interact with theLangevin thermostat. The intensity of heat exchangewith the thermostat is characterized by a relaxation time t r . The value t r = 1 ps was used in the simulation. In thetime t = 100 t r the sheet being fully thermalized. Theanalysis of the further dynamics of the sheet allows us tofind the average temperature of the circular membrane T m = 13 N m k B N X n =1 (1 − δ n ) M n h ( ˙ u n , ˙ u n ) i , N m = N X n =1 (1 − δ n ) , where summation occurs only for atoms not in contactwith the substrate ( N m is the number of such atoms), and the average value h ( ˙ u n , ˙ u n ) i = lim t →∞ t Z t + tt ( ˙ u n ( τ ) , ˙ u n ( τ )) dτ . When the substrate is stationary (when the oscillationamplitude A = 0), the temperature of the membraneis always equal to the temperature of the thermostat( T m = T ). Therefore, additional thermalization of themembrane can be characterized by a temperature differ-ence ∆ T = T m − T .Let us take the oscillation amplitude A = 1, 2 ˚A · cm − ,the temperature of the thermostat T = 300K. The depen-dence of the additional thermalization of the membrane∆ T on the frequency of vertical oscillations of the sub-strate ω is shown in Fig. 4. As can be seen from thefigure, the additional thermalization of the membrane isdifferent from zero only near certain frequency values, thenumber of which increases with increasing membrane ra-dius (see Fig. 5). Because by the vertical displacementof the substrate on all the edge atoms of the circularmembrane are the same forces, the vertical vibrations ofthe substrate in the membrane can only cause vibrationswith circular symmetry. Therefore, additional thermal-ization occurs only at frequencies close to the frequenciesof the intrinsic nodeless oscillations of the membrane,which have a circular symmetry (the amplitude of thedisplacements of the membrane atom depends only onits distance from the center of the membrane). Thus,additional thermalization of the membrane occurs pri-marily due to the resonant pumping of its own circularlysymmetric oscillations.A similar resonant pumping of the membrane eigen-modes occurs for the graphene and h-BN sheets for allconsidered substrates – see Fig. 5. As can be seen fromthe figure, the resonant pumping of the main oscillationoccurs almost equally for all membranes. The differ-ences appear only for higher frequency resonances. Asthe membrane radius increases, the resonance frequenciesdecrease and their number increases. The analysis of theenergy distribution of resonant vibrations of the mem-brane (see Fig. 6) shows that the distribution always hasa circular symmetry and has several nodal circles whosenumber coincides with the number of the resonance fre-quency. This shows that resonance pumping occurs pri-marily due to the excitation of natural oscillations of themembrane with circular symmetry (i.e. oscillations hav-ing only nodal circles).Let us consider in more detail the first resonance of themembrane. As can be seen in Fig. 4 and 5, when the fre-quency increases, the vibrational energy of the membraneinitially grows monotonically, at a certain frequency ω r reaches its maximum value, and then sharply decreases tothe background value of the energy of thermal vibrations.Therefore, it is convenient to determine the frequency ofthe first resonance as the average value¯ ω = 1 C Z . ω r ω ∆ T ( ω ) dω, C = Z . ω r ∆ T ( ω ) dω. Figure 6: Temperature distribution in a circular graphenemembrane of radius R = 6 nm (h-BN substrate, amplitudeof forced substrate oscillations A = 2˚A · cm − ) at: (a) firstresonance (frequency ω = 1 . − , maximum temperature T m = 331K); (b) second resonance ( ω = 5 . − , T m =324K); (c) third resonance ( ω = 9 . − , T m = 324K); (d)the fourth resonance ( ω = 16 . − , T m = 366K). Blue colorcorresponds to the background temperature T = 300K, redcolor corresponds to the maximum temperature T m . Similarly, we can define the frequencies of next reso-nances ¯ ω i , i = 2, 3, ....The results of numerical simulation of membrane vi-brations are shown in Fig. 4. The figure shows that each i -th eigen membrane vibration with a circular (radial)symmetry corresponds to resonant membrane vibrationwith frequency ¯ ω i > ω i . The resonance frequency is al-ways higher than the frequency of the corresponding nat-ural membrane vibration but lower than the frequency ofthe next natural vibration: ω i < ¯ ω i < ω i +1 , i = 1, 2, 3 ,.... The larger the amplitude A of forced substrate vibra-tion gets, the stronger the resonance frequency shifts tothe right. This indicates the nonlinearity of resonancesdue to rigid anharmonicity of membrane natural vibra-tion at high energy (the frequency of natural vibrationincreases with increasing vibration amplitude).The analysis of dependency of the resonance frequency¯ ω i on membrane radius R shows that as the radius in-creases, the resonance frequency decreases slower thanthe frequency of the corresponding natural membrane vi-bration ω i : ¯ ω i ∼ d i / ( R + R i ) α i , α i < , (11) ln( R ) l n ( ω ) Figure 7: Dependence of oscillation frequencies ω i ( i = 1, 2,3, markers 1, 2, 3) and resonance frequencies ¯ ω i on membraneradius R for graphene on h-BN substrate. Blue curves 1, 2,3 give approximations ω i = c i / ( R + R i ) , c i = 3950, 15500,34500 cm − · ˚A , R i = 4 .
0, 4.7, 5.0 ˚A, i = 1, 2, 3. Markers 4give resonance frequencies ¯ ω i for amplitude of forced substratevibrations A = 1, markers 5 – for A = 2 ˚A · cm − . Greencurves give approximations ¯ ω i = d i / ( R + R i ) α i for A = 1( d i = 1650, 11900, 32000, α i = 1 .
7, 1.9, 1.96), red curves –for A = 2 ( d i = 1270, 10500, 2800 cm − · ˚A α i , α i = 1 .
6, 1.85,1.92). Dimension of the frequency [ ω ] =cm − , radius [ R ] =˚A. – see Fig. 7. The greater the amplitude A of the substrateoscillation, the lower the value of exponent α i . For thefirst resonance ( i = 1) the exponent α = 1 . A = 1,and α i = 1 . A = 2 ˚A · cm − . The deceleration of thedecrease of the resonance frequencies ¯ ω i with increasingradius R is caused by the anharmonicity of the membranevibrations. VI. CONCLUSIONS
We have simulated natural and resonant oscillations ofsuspended circular graphene and hexagonal boron nitride(h-BN) membranes using full-atomic models. The pres-ence of the substrate (of flat surface of graphite and h-BN crystal, hexagonal ice, silicon carbide 6H-SiC(0001),nickel surface (111)) leads to the forming of a gap at thebottom of the frequency spectrum of transversal vibra-tions of the sheet. Frequencies of natural oscillations ofthe membrane ω i always lie in this gap, and they de-crease with the increasing radius of the membrane R as ( R + R i ) − with nonezero effective increase of ra-dius R i >
0. The modeling of the sheet dynamics hasshown that small periodic transversal displacements ofthe substrate lead to resonant vibrations of the mem-branes, at frequencies close to the eigenfrequencies ofnodeless vibrations of the membranes with circular sym-metry. The energy distribution of the resonant vibrationsof the membrane has a circular symmetry and severalnodal circles whose number coincides with the numberof the resonant frequency i . The frequencies of the reso-nances decrease by increasing the radius of the membraneas ( R + R i ) α i with exponent α i <
2. The lower rate ofthe resonance frequency decrease is caused by the anhar-monicity of membrane vibrations.
Acknowledgements
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